# Full text of "Treatise On Analysis Vol-Ii"

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```348       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

Let (xn)n^1 be a dense sequence in H. We define the Hn inductively, as
follows. H! is the closure of the vector subspace of H generated by the
U(s) • xl9 as s runs through A; we have xl e Hl9 because A has a unit element.
If H!, ...,!!„_! have been constructed, it may be the case that H is equal to
the (direct) sum L of the Hf, and the construction stops. If not, let U be the
orthogonal supplement of L in H, and let p(n) be the smallest integer such
that, if x'n is the orthogonal projection of xp^n) on L', the vector subspace
H^ of L' generated by the U(s) - x'n is not equal to {0}: there exists such an
index because the representation Uis nondegenerate, and we take Hn to be the
closure of H^. Since x'neHn, the subspaces Hn satisfy the required conditions,
by virtue of (6.4.2).

(15.5.7) If A is a Banach algebra with involution, having a unit element, then
every representation s*-+U(s) of A in a Hilbert space satisfies \\U(s)\\ <£ \\s\\
(and in particular C/is a continuous mapping of A into «Sf(H)).

We have \\U(s)\\2 = \\U(s)*U(s)\\ = p(J7(j)*Z7(j)) in the star algebra j?(H)
(15.4.14.1). Since U(sy*U(s) = U(s*s), it follows from (15.2.8Q) that

In particular, we recover in this way (15.3.1(ii)).

6. POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS
AND REPRESENTATIONS

Let A be an algebra with involution (not necessarily normed, and not
necessarily having a unit element). A linear form/: A -*• C on A is said to be
positive if

(15.6.1)'                                   /(**jc)^0

for all jc e A.

(15.6.2)   Let fbe a positive linear form on an algebra A with involution.

(i) The mapping (x, y)*-+ff(x9 y) =f(y*x) is a positive hermitian form on
A x A: in other words,

(15.6.2.1)                             /(**J>) =/(/**)

for all x, y in A.ns through A, is dense in H. (In any case, this subspace is stable with
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